Automorphic Representations and L-Functions for the General Linear Group: Volume 1

This graduate-level textbook provides an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. Definitions are kept to a minimum and repeated when reintroduced so that the book is accessible from any entry point, and with no prior knowledge of representation theory. The book includes concrete examples of global and local representations of GL(n), and presents their associated L-functions. In Volume 1, the theory is developed from first principles for GL(1), then carefully extended to GL(2) with complete detailed proofs of key theorems. Several proofs are presented for the first time, including Jacquet's simple and elegant proof of the tensor product theorem. In Volume 2, the higher rank situation of GL(n) is given a detailed treatment. Containing numerous exercises by Xander Faber, this book will motivate students and researchers to begin working in this fertile field of research.

Preface; 1. Adeles over Q; 2. Automorphic representations and L-functions for GL(1,AQ); 3. The classical theory of automorphic forms for GL(2); 4. Automorphic forms for GL(2,AQ); 5. Automorphic representations for GL(2,AQ); 6. Theory of admissible representations of GL(2,Qp); 7. Theory of admissible (g,K8) modules for GL(2,R); 8. The contragredient representation for GL(2); 9. Unitary representations of GL(2); 10. Tensor products of local representations; 11. The Godement–Jacquet L-function for GL(2,AQ); Solutions to selected exercises; References; Symbols index; Index.

This modern approach to the theory of automorphic representations keeps definitions to a minimum, focusing instead on providing concrete examples and detailed proofs of the key theorems. This book is the perfect introduction for students at the advanced undergraduate level and beyond, and for researchers new to the field.

Dorian Goldfeld is a Professor in the Department of Mathematics at Columbia University, New York.
Joseph Hundley is an Assistant Professor in the Department of Mathematics at Southern Illinois University, Carbondale.